VINZONS PILOT HIGH SCHOOL
QUARTER 3
ADVANCED MATHEMATICS
Prepared by: BabylynClacioDatio2023
First name:
·
Date: ____________________ ·
Section: ______________________
Last name:
·
Score: _________ /30
Signature:
READ each question carefully. Use blue or black pen only. Good luck! ☺
PART I: Complete the mathematical induction below.
1. Prove using mathematical induction that for all 𝑛 ≥ 1,
1 + 4 + 7 +∙∙∙ +(3𝑛 − 2) =
Solution:
For any integer 𝑛 ≥ 1, let 𝑃𝑛 be the statement that
1 + 4 + 7 +∙∙∙ +(3𝑛 − 2) =
Base Case. The statement 𝑃1 says that
1=
𝑛(3𝑛 − 1)
.
2
𝑛(3𝑛 − 1)
2
1(3 − 1)
,
2
which is true.
Inductive Step. Fix 𝑘 ≥ 1, and suppose that 𝑃𝑘 holds, that is,
𝑘(3
1 + 4 + 7 +∙∙∙ +(3𝑘 − 2) =
It remains to show that 𝑃𝑘+1 holds, that is,
− 1)
2
.
(𝑘 + 1)(3(𝑘 + 1) − 1)
.
2
1 + 4 + 7 +∙∙∙ +(3(𝑘 + 1) − 2) = 1 + 4 + 7 +∙∙∙ +(3(𝑘 + 1) − 2)
)
= 1 + 4 + 7 +∙∙∙ +(
𝑘(3𝑘 − 1)
1 + 4 + 7 +∙∙∙ +(3𝑘 − 2) + (3𝑘 + 1) =
+ (3𝑘 + 1)
2
𝑘(3𝑘 − 1) + (3𝑘 + 1)
=
2
3𝑘 2 − 𝑘 +
+2
=
2
3𝑘 2 + 5𝑘 + 2
=
2
=
1 + 4 + 7 +∙∙∙ +(3(𝑘 + 1) − 2) =
=
(𝑘 + 1)(3(𝑘 + 1) − 1)
2
Therefore, 𝑃𝑘+1 holds.
Thus, by the principle of mathematical induction, for all 𝑛 ≥ 1, 𝑃𝑛 holds.
PART II: Solving linear equations. (5 points each)
1. Solve for the unknowns. BOX your final answer.
{
Using Elimination Method
Solution:
5𝑥 + 6𝑦 = 7
3𝑥 + 4𝑦 = 5
Using Gaussian Elimination Method (Augmented Matrix)
Solution:
PART III: Find the value of the variables.
2
1. [
𝑎
𝑒
3
] + 𝑘[
3
0
1
8
6
]=[
]
−2
−3 −1
2
2. [ 𝑤
−3
Find: 𝑎 =_______, 𝑒 =_______, 𝑘 =_______
𝑦
3
6
1
𝑦
−2 3 −1
−1] + [ 3 𝑥 6 ] = [7
𝑤 2 1
𝑥
1
3 𝑦
8 𝑥]
8 6
Find: 𝑤 =_______, 𝑥 =_______, 𝑦 =_______
PART IV: Are these matrices compatible for multiplication? If so, what is the size of the product matrix?
1) (
2 −1 0
)(
3 4
1
5
)
−2
3)
Yes_____ No______
Size: _____________
2) (
4
1 5 1
) (1 )
4 −6 1
3
(
Yes_____ No______
Size: _____________
2 −1 3
)( )
3 4
1
4)
Yes_____ No______
Size: _____________
1 −3
4 3
)( 0
1)
1 2
−7 0
(
Yes_____ No______
Size: _____________
5) In the expression 𝐴 × 𝐵, if 𝐴 is 5 × 6 matrix then what could be the dimensions of 𝐵?
𝐵 = _______________________
PART V: Refer on the given matrices below.
−3
𝐴=[ 2
−1
Find: (5 points each)
1. 𝐴 × 𝐵 =
2. 𝐴(𝐵 + 𝐶) =
1
2]
5
−1 1
𝐵=[
2 5
−1 3
]
−3 1
𝐶=[
3 2
1 6
−2 1
]
−2 4